Group of integers modulo n

Definition

Let be a positive integer. The group of integers modulo is an Abelian group defined as follows:

Its underlying set is the set

The rule for addition in the group is as follows. If the integer sum is between and , then the sum is defined as equal to the integer sum. If the integer sum is at least , then the sum is defined as .

The identity element of the group is .

The inverse map in the group is defined as follows: the additive inverse of is , and the additive inverse of any other is, as an integer, .

The group of integers modulo is a concrete description of the cyclic group of order .

This group is typically denoted as or simply . It is also sometimes denoted as .

Examples

Here are the multiplication tables (more aptly called addition tables, because the group is Abelian and the operation is more typically called addition) for the group of integers mod for small values of :